# Projection Robust Wasserstein Barycenters

arXiv: https://arxiv.org/abs/2102.03390 · HTML: https://mlsanity.com/paper/2102.03390/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 32 references that are in the map.
- First posted on arXiv 5 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Fast Discrete Distribution Clustering Using Wasserstein Barycenter with Sparse Support](https://mlsanity.com/paper/1510.00012/) (2015)
- [Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm](https://mlsanity.com/paper/2002.04783/) (2020)
- [Wasserstein barycenters are NP-hard to compute](https://mlsanity.com/paper/2101.01100/) (2021)
- [A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance](https://mlsanity.com/paper/2012.05199/) (2020)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Interior-Point Methods Strike Back: Solving the Wasserstein Barycenter Problem](https://mlsanity.com/paper/1905.12895/) (2019)
- [On the Computational Complexity of Finding a Sparse Wasserstein Barycenter](https://mlsanity.com/paper/1910.07568/) (2019)

## Built on by (most cited first, 8 of 8)

- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Efficiently Escaping Saddle Points in Bilevel Optimization](https://mlsanity.com/paper/2202.03684/) (2022)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances](https://mlsanity.com/paper/2405.15441/) (2024)
- [On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport](https://mlsanity.com/paper/2109.15030/) (2021)
- [Spectral Clustering for Discrete Distributions](https://mlsanity.com/paper/2401.13913/) (2024)
- [Generalized infinite dimensional Alpha-Procrustes based geometries](https://mlsanity.com/paper/2511.09801/) (2025)

## Abstract (arXiv)

Collecting and aggregating information from several probability measures or histograms is a fundamental task in machine learning. One of the popular solution methods for this task is to compute the barycenter of the probability measures under the Wasserstein metric. However, approximating the Wasserstein barycenter is numerically challenging because of the curse of dimensionality. This paper proposes the projection robust Wasserstein barycenter (PRWB) that has the potential to mitigate the curse of dimensionality. Since PRWB is numerically very challenging to solve, we further propose a relaxed PRWB (RPRWB) model, which is more tractable. The RPRWB projects the probability measures onto a lower-dimensional subspace that maximizes the Wasserstein barycenter objective. The resulting problem is a max-min problem over the Stiefel manifold. By combining the iterative Bregman projection algorithm and Riemannian optimization, we propose two new algorithms for computing the RPRWB. The complexity of arithmetic operations of the proposed algorithms for obtaining an ε-stationary solution is analyzed. We incorporate the RPRWB into a discrete distribution clustering algorithm, and the numerical results on real text datasets confirm that our RPRWB model helps improve the clustering performance significantly.